Mathematical Tool of Quantum Mechanics | Quantum Science Philippines
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Mathematical Tool of Quantum Mechanics



Consider the set of all entities of the form (a, b, c) where the entries are real numbers. Addition and scalar multiplication are defined as follows: 

\begin{array}{ccc} (a, b, c) + (d, e, f) &=& (a+d, b+e, c+f)\\ a(a, b, c) &=& (aa, ab,ac) \end{array}

i.) Write down the null vector and inverse of (a, b, c).

ii.) Show that vectors of the form (a, b, 1) do not form a vector space.



i.) Let (x, y, z) be the null vector of (a, b, c), such that

(a, b, c) + (x, y, z) = (a, b, c)

Then as defined,

(a+x, b+y, c+z) = (a, b, c)

and by comparison,

\begin{array}{ccc} a+x &=& a \\ b+y &=& b \\ c+z &=& c \end{array}

If we are to solve these equations for the values of x, y and z, we could get

\begin{array}{ccc} x &=& 0 \\ y &=& 0 \\ z &=& 0 \end{array}.

Therefore, the null vector is given as

\mathbf{(x, y, z)} = \mathbf{(0, 0, 0)}.


Now, we let the inverse of  (a, b, c) be  (\alpha, \beta, \gamma), such that following the definition of an inverse vector we have

(a, b, c) + ( \alpha , \beta , \gamma) = (0, 0, 0)

( a + \alpha , b + \beta , c+ \gamma ) = (0, 0, 0)

Following the same process as before, if we compare we can obtain

\begin{array}{ccc} a+\alpha &=& 0 \\ b+\beta &=& 0 \\ c+\gamma &=& 0 \end{array}

Solving for \alpha, \beta and \gamma,

\begin{array}{ccc} \alpha &=& -a \\ \beta &=& -b \\ \gamma &=& -c \end{array}.

Therefore, the inverse vector of (a, b, c) is

\mathbf{( \alpha, \beta, \gamma )} = \mathbf{( -a, -b, -c)}.



ii.) The vector  (a, b, 1) do not form a vector space because:

a.) it violates the closure under addition,

\begin{array}{ccc} ( a, b, 1) + ( c, d, 1) &=& ( a+c, b+d, 1+1) \\ &=& \mathbf{( a+c, b+d, 2) \not\in (a,b, 1)} \end{array}.

               b.) it violates the closure under scalar multiplication,

\lambda (a, b, 1) = \mathbf{( \lambda a, \lambda b, \lambda ) \not\in ( a, b, 1)}.

               c.) there is NO null vector

(0, 0, 0) \not\in (a, b, 1)

               d. an additive inverse does not exist

( -a, -b, -1) \not\in (a, b, 1).



By: Joshua J. Ordeniza, MS Physics Student, MSU-IIT



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